bioRxiv preprint doi: https://doi.org/10.1101/824847; this version posted October 30, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. Structure of the space of taboo-free sequences Cassius Manuel · Arndt von Haeseler . Abstract Models of sequence evolution typically assume that all sequences are possible. However, restriction enzymes that cut DNA at specific recogni- tion sites provide an example where carrying a recognition sequence can be lethal. Motivated by this observation, we studied the set of strings over a finite alphabet with taboos, that is, with prohibited substrings. The taboo-set is referred to as T and any allowed string as a taboo-free string. We consider the graph Γn(T) whose vertices are taboo-free strings of length n and whose edges connect two taboo-free strings if their Hamming distance equals 1. Any (random) walk on this graph describes the evolution of a DNA sequence that avoids deleterious taboos. We describe the construction of the vertex set of Γn(T). Then we state conditions under which Γn(T) and its suffix subgraphs are connected. Moreover, we provide a simple algorithm that can determine, for an arbitrary T, if all these graphs are connected. We concluded that bac- terial taboo-free Hamming graphs are nearly always connected, although 4 properly chosen taboos are enough to disconnect one of its suffix subgraphs. Austrian Science Fund (FWF, grant number I-1824-B22). Corresponding author: Cassius Manuel ORCID ID: https://orcid.org/0000-0002-6297-4762 Center for Integrative Bioinformatics Vienna, Max Perutz Labs University of Vienna, Medical University of Vienna Dr. Bohr Gasse 9, A-1030 Vienna, Austria E-mail: [email protected] Arndt von Haeseler ORCID ID: https://orcid.org/0000-0002-3366-4458 Center for Integrative Bioinformatics Vienna, Max Perutz Labs University of Vienna, Medical University of Vienna Dr. Bohr Gasse 9, A-1030 Vienna, Austria Faculty of Computer Science, University of Vienna W¨ahringerStr. 29, A-1090 Vienna, Austria. E-mail: [email protected] bioRxiv preprint doi: https://doi.org/10.1101/824847; this version posted October 30, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 2 Cassius Manuel, Arndt von Haeseler Keywords Bacteriophage DNA evolution · Endonuclease-dependent evo- lution · Restriction-enzyme dependent evolution · Restriction-modification system · Hamming graph with taboos · Connectivity of Hamming graphs Mathematics Subject Classification (2010) 05C40 · 92D15 1 Introduction In bacteria, restriction enzymes cleave foreign DNA to stop its propagation. To do so, a double-stranded cut is induced by a so-called recognition sequence, a DNA sequence of length 4 − 8 base pairs ([2]). As part of their recognition- modification system, bacteria can escape the lethal effect of their own restric- tion enzymes by modifying recognition sequences in their own DNA ([14]). Nevertheless, a significant avoidance of recognition sequences has been ob- served in bacterial DNA ([7], [19]). Also in bacteriophages, the avoidance of the recognition sequences is evolutionary advantageous ([19]). Therefore the recognition sequence is, as we call it, a taboo in a majority of fragments of both host and foreign DNA. Motivated by this biological phenomenon, we studied the Hamming graph Γn(T), whose vertices are strings of length n over a finite alphabet Σ not containing any taboo of the set T as substring. By definition, an edge connects two strings of this graph if their Hamming distance is 1, i.e. if they differ by a single substitution. Moreover, given a taboo-free string s, we consider s the subgraph Γn(T) of Γn(T) induced by every taboo-free string with suffix s. Suffix s represents a conserved DNA fragment, that is, a sequence which remained invariable during evolution ([20], [6]). Instead of studying just the connectivity of Γn(T), here we define conditions s under which every graph of the form Γn(T) is connected. Note that this result e is stronger than just the connectivity of Γn(T), for Γn(T) = Γn(T), where e is the empty string. If taboo-set T satisfies that, for every taboo-free string s s and integer n, graph Γn(T) is connected, then evolution can explore all the space of taboo-free sequences by simple point mutation, no matter which DNA suffix fragments remain invariable. Notice that we assume that the taboo-set T does not change in the course of evolution. 2 Spaces of taboo-free sequences We start with a non-biological example of a space of taboo-free strings. For the binary alphabet Σ = f0; 1g and T = f11g, graphs Γn(T) are known as Fibonacci cubes, proposed in [9] and [10] as a network topology in parallel computing. Fibonacci cubes are connected, bipartite and contain a Hamilto- nian path. Enumeration results regarding e.g. the Wiener Index or the degree distribution have been obtained, mainly using generating functions, in [13], [5] and [16], among others (see the compendium [12]). bioRxiv preprint doi: https://doi.org/10.1101/824847; this version posted October 30, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. Structure of the space of taboo-free sequences 3 Other types of taboo strings assuming a binary alphabet Σ = f0; 1g have been considered in [10] for T = f1 ::: 1g, while the case T = fsg for an arbitrary binary string s was studied in [11]. The number of strings over any finite alphabet with taboos has been studied, using generating functions, in [8], while related software is given in [17]. Therefore our terminology will be based on this previous work: From now on we will use the terms string to refer to a sequence of symbols over an alphabet Σ, while (DNA) sequence is reserved for biological contexts, where the alphabet consists of the four nucleotides, i.e. Σ = fA; C; G; T g . Given a string s, its length is denoted by jsj. Given two strings s1; s2 of equal length, then d(s1; s2) denotes their Hamming distance, that is, the number of sites at which the corresponding symbols differ. Given a finite set of strings T, we call T a taboo-set if, for any t 2 T, jtj≥ 2. We refer to the strings of T as taboos. The length of the longest taboo in T will be denoted by M := max fjtjgt2T.A taboo-free string is a string not having any taboo of T as substring. The set of taboo-free strings of length s n is denoted by Vn(T). Given a taboo-free string s, for n ≥ jsj, Vn (T) denotes the set of strings of Vn(T) with suffix s. The Hamming graph of taboo-free strings of length n, denoted by Γn(T) := (Vn(T); En(T)), is the graph with vertex set Vn(T) such that two vertices u; v 2 Vn(T) are adjacent if their Hamming distance equals 1. See Fig. s 1 for an example where Γn(T) is disconnected for n ≥ 3. Analogously, Γn(T) s is the Hamming graph with vertex set Vn (T). Fig. 1 Graph Γn(T) for n 2 [1; 5] for binary alphabet Σ = f0; 1g and T = f11; 000g. Set Vn+1(T) is constructed by adding every allowed symbol at the beginning of each string in Vn(T). Taboo-sets as generated by the avoidance of restriction sequences can assume various levels of complexities. We discuss some examples from RE- BASE ([18]). Note that many restriction enzymes of this database have an unknown recognition sequence, hence our taboo-sets may underestimate the actual amount of taboos. Before studying the examples, we will briefly review essential nomenclature for DNA sequences. bioRxiv preprint doi: https://doi.org/10.1101/824847; this version posted October 30, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 4 Cassius Manuel, Arndt von Haeseler DNA is double-stranded, where A pairs with T and G pairs with C, thus it suffices to discuss only one of the strands. We adopt the convention that, given any of the strands, the DNA sequence is always represented from the 5' end to the 3' end (which is chemically determined). As a consequence, given a DNA sequence, its complementary DNA sequence, the one lying on the opposite strand, is obtained by inverting the order of the symbols and carrying through substitutions A $ T and C $ G. If a DNA sequence s is identical to its complementary DNA sequence, we say that s is an inverted repeat (see [22]). For example, sequence CCGG is an inverted repeat. The fact that DNA is double-stranded implies that each recognition se- quence induces taboos in pairs, namely itself and its complementary DNA sequence. For example, if AGGGC is a recognition sequence, then also the complementary strand GCCCT is a taboo. If, however, the recognition se- quence is an inverted repeat such as T GCA, then this pair is actually one single recognition sequence. This is the case in most of type II restriction- modification systems ([7]). 2.1 Turneriella parva The Turneriella parva (REBASE organism number 8970) strain produces a restriction enzyme with recognition sequence GAT C, an inverted repeat. Simi- larly, another of its enzymes has recognition sequences GGACC and GGT CC. Thus, these restriction enzymes generate the taboo-set [ TT:pa = fGAT Cg fGGACC; GGT CCg: (1) 2.2 Helicobacter pylori In H.